What are some applications where tree models perform better than neural networks?
Hard to say in general. Speaking from my own experience and by looking at which models win Kaggle competitions (see here and here), I would say tree-based models e.g. Random Forests, Decision Trees, Gradient Boosting are favorable over neural networks when working with low-dimensional data and easy interpretable features (usually simple tabular data with numeric, ordinal or categorical features).
Whereas when working with everything high-dimensional like images, text, time series or other data with non-trivial features, I would recommend neural networks.
Of course there might be exceptions and the future may prove me wrong.